OCR C3 (Core Mathematics 3) 2010 June

Find \(\frac{dy}{dx}\) in each of the following cases:

1. \(y = x^3 e^{2x}\), [2]
2. \(y = \ln(3 + 2x^2)\), [2]
3. \(y = \frac{x}{2x + 1}\). [2]

The transformations R, S and T are defined as follows. \begin{align} \text{R} &: \text{ reflection in the } x\text{-axis}
\text{S} &: \text{ stretch in the } x\text{-direction with scale factor 3}
\text{T} &: \text{ translation in the positive } x\text{-direction by 4 units} \end{align}

1. The curve \(y = \ln x\) is transformed by R followed by T. Find the equation of the resulting curve. [2]
2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]

1. Express the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\) in the form \(a \sin^2 \theta + b \sin \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Hence solve, for \(-180° < \theta < 180°\), the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\). [3]

\includegraphics{figure_4} The diagram shows part of the curve \(y = \frac{k}{x}\), where \(k\) is a positive constant. The points A and B on the curve have \(x\)-coordinates 2 and 6 respectively. Lines through A and B parallel to the axes as shown meet at the point C. The region R is bounded by the curve and the lines \(x = 2\), \(x = 6\) and \(y = 0\). The region S is bounded by the curve and the lines AC and BC. It is given that the area of the region R is \(\ln 81\).

1. Show that \(k = 4\). [3]
2. Find the exact volume of the solid produced when the region S is rotated completely about the \(x\)-axis. [4]

1. Solve the inequality \(|2x + 1| \leqslant |x - 3|\). [5]
2. Given that \(x\) satisfies the inequality \(|2x + 1| \leqslant |x - 3|\), find the greatest possible value of \(|x + 2|\). [2]

1. Show by calculation that the equation $$\tan^2 x - x - 2 = 0,$$ where \(x\) is measured in radians, has a root between 1.0 and 1.1. [3]
2. Use the iteration formula \(x_{n+1} = \tan^{-1}\sqrt{2 + x_n}\) with a suitable starting value to find this root correct to 5 decimal places. You should show the outcome of each step of the process. [4]
3. Deduce a root of the equation $$\sec^2 2x - 2x - 3 = 0.$$ [3]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = (3x - 1)^4\). The point P on the curve has coordinates \((1, 16)\) and the tangent to the curve at P meets the \(x\)-axis at the point Q. The shaded region is bounded by PQ, the \(x\)-axis and that part of the curve for which \(\frac{1}{3} \leqslant x \leqslant 1\). Find the exact area of this shaded region. [10]

1. Express \(3 \cos x + 3 \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [3]
2. The expression T\((x)\) is defined by T\((x) = \frac{8}{3 \cos x + 3 \sin x}\).
   1. Determine a value of \(x\) for which T\((x)\) is not defined. [2]
   2. Find the smallest positive value of \(x\) satisfying T\((3x) = \frac{8}{3}\sqrt{6}\), giving your answer in an exact form. [4]

The functions f and g are defined for all real values of \(x\) by $$f(x) = 4x^2 - 12x \quad \text{and} \quad g(x) = ax + b,$$ where \(a\) and \(b\) are non-zero constants.

1. Find the range of f. [3]
2. Explain why the function f has no inverse. [2]
3. Given that \(g^{-1}(x) = g(x)\) for all values of \(x\), show that \(a = -1\). [4]
4. Given further that gf\((x) < 5\) for all values of \(x\), find the set of possible values of \(b\). [4]